A function $f:\Bbb R \rightarrow \Bbb R$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers $\alpha_0, \ldots, \alpha_{n-1}$ such that $s^{n+1}\int_0^\delta e^{-st}[f(x+t)-\sum_{i=0}^{n-1}\alpha_i t^i/i!]\,dt$ converges as $s\rightarrow +\infty$ for some $\delta>0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }(x)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized Peano derivative; hence the Laplace derivative generalizes the Peano and ordinary derivatives.
@article{119247,
author = {Ralph E. Svetic},
title = {The Laplace derivative},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {331-343},
zbl = {1051.26004},
mrnumber = {1832151},
language = {en},
url = {http://dml.mathdoc.fr/item/119247}
}
Svetic, Ralph E. The Laplace derivative. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 331-343. http://gdmltest.u-ga.fr/item/119247/
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